Abraham Lincoln (1930) [Abraham Lincoln]: Shot in the head by Ian Keith while Walter is sitting in the balcony in the theatre.

The Beast of the City(1932) [Captain James Fitzpatrick]: Shot in the chest during the big shoot-out with Jean Hersholt's thugs at Jean's club; he dies shortly after all the gangsters have been wiped out. (Thanks to McQuade)

The Maltese Falcon (1941) [Captain Jacobi]: Shot (off-screen) by Elisha Cook Jr.; he manages to carry the Maltese Falcon to Humphrey Bogart's office before he collapses on the couch and dies. (Thanks to Betty)

Yankee Doodle Dandy (1942) [Jerry Cohan]: Dies of natural causes/old age, with his son (James Cagney) at his bedside. (Thanks to Betty)

And Then There Were None (1945) [Dr. Edward G. Armstrong]: Pushed off of a cliff (off-screen) by Barry Fitzgerald; his body is shown afterwards when the others discover him. (Thanks to PortsGuy)